The fact that "there are more Pryms than Jacobians" can be used to establish structural results on the moduli space of abelian varieties.
For instance, in dimension 5, the numbers \(\binom{g+1}{2}\)  and \(3g\)  coincide, so Prym varieties are full-dimensional in \(\mathcal {A}_5\) . A close examination of the geometry of Prym varieties then leads to the conclusion that the space \(\mathcal {A}_5\)  is uniruled [1]}.
On a different note, the intermediate Jacobian of a smooth Fano 3-fold \(X\)  is a Prym variety [2]}, and \(X\)  is rational iff it is an actual Jacobian.
Clemens and Griffiths showed that this Prym variety is never a Jacobian when \(X\)  is a smooth cubic threefold, thus proving that smooth cubic threefolds are non-rational [3]}.
The fact that intermediate Jacobians are Prym varieties is also used by Sacca, Laza, and Voisin in [4]} to construct and compactify hyperkähler manifolds.
